Turn: ($0.85) 4(2 players)MP1 checks, Hero bets $0.61, MP1 raises to $0.72, Hero calls $0.11.I have no idea how to play it right. Perhaps I am beat, but still there are plenty of overpairs and draws that I beat.

You can also 3-bet JJ. It is strong enough against midstacks who might easily open with worse hands. I like your postflop raise and I would play for stacks, since turn is not the worst hand I would also put MP1 all-in